Finite Group Databasebeta

← #27281619

Subgroups — #27281619

Order 13568.

540 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285168C211yesmaximal, characteristic
2#20273428C212yesmaximal, complemented
3#20273350C211yesmaximal, characteristic, complemented
4#20272586C212yesmaximal, complemented
5C53×#2980C211yesmaximal, characteristic, complemented
6C2×#13133242—22no—
7C2×#13133214C2211yescharacteristic
8#16811013—22no—
9#16810999C412yes—
10#16810997C412yes—
11#16810680C2212yes—
12C4×#11719917C2211yescharacteristic
13#16810366—24no—
14#16810363—22no—
15C53×#532C2211yescharacteristic
16C53×#435—24no—
17C53×#432C2212yes—
18C53×#408—22no—
19C22×D212—21no—
20C22×D212—22no—
21#13133242—44no—
22#13133242—44no—
23C2×#11719924—24no—
24C2×#11719922—24no—
25C2×#11719917—21no—
26#13133214—21no—
27#13133214—22no—
28#13133214C2×C412yes—
29C4⋊#88322—22no—
30C22⋊#88322—22no—
31#13133128—21no—
32#13133128—22no—
33C4×#88322C2×C412yes—
34C2×#41×C53D412yes—
35C2×M16×C53—22no—
36C22×C8×C53—22no—
37C4⋊Q8×C53C2×C411yescharacteristic
38#127×C53C2×C411yescharacteristic
39#124×C53D412yes—
40#121×C53—22no—
41C4×D4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53D414yes—
44C22⋊C8×C53—24no—
45C22⋊C8×C53—22no—
46#97×C53—22no—
47C23×D53—41no—
48#41×C53—42no—
49#41×C53—42no—
50C2×Q8×C53C22⋊C412yes—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53C22⋊C412yes—
54C22×C4×C53—22no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4112yes—
57C2×D212—22no—
58C2×D212—24no—
59C2×D212—42no—
60C2×D212—42no—
61C2×D212—41no—
62C2×D212—42no—
63C2×D212—84no—
64M16×C53—82no—
65C2×C8×C53—24no—
66C2×C8×C53—82no—
67C4⋊C4×C53—22no—
68C22⋊C4×C53—24no—
69C42×C53C22×C411yescharacteristic
70C42×C53C2×D411yescharacteristic
71C42×C53C2×D412yes—
72#11719924—48no—
73#11719922—48no—
74#11719917—24no—
75C2×#88322—22no—
76C2×#88322—42no—
77C23×C53—22no—
78C23×C53#12311yescharacteristic
79C22×D53—42no—
80C22×D53—81no—
81C22×D53—81no—
82C22×D53—84no—
83Q8×C53—42no—
84D4×C53—42no—
85D4×C53—82no—
86D4×C53—82no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53C4×D412yes—
94C2×C4×C53#12011yescharacteristic
95C2×C4×C53#12112yes—
96D212—44no—
97D212—84no—
98D212—84no—
99D212—82no—
100C8×C53—44no—
101#88322—24no—
102#88322—82no—
103#9423—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—22no—
106C22×C53—42no—
107C22×C53—44no—
108C22×C53#38611yescharacteristic
109C2×D53—82no—
110C2×D53—84no—
111C2×D53—162no—
112C2×D53—162no—
113C4×C53—21no—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—42no—
117C4×C53#45312yes—
118#3151—531no—
119#2980D5311yescharacteristic, complemented
120#1555—532no—
121#1536—531no—
122#1291—532no—
123C2×C53—41no—
124C2×C53—82no—
125C2×C53#188711yescharacteristic
126C2×C53#190312yes—
127D53—162no—
128C2×AGL(1,ℤ/8)—1062no—
129#532C2×D5311yescharacteristic
130C2×#127—531no—
131#496—1062no—
132#493—532no—
133#492—532no—
134#435—24no—
135#432—532no—
136#432C2×D5312yes—
137#408—22no—
138C4×M16—531no—
139#331—1064no—
140#329—1062no—
141C53#942311yescharacteristic, complemented, Sylow
142C2×#41C22⋊D5312yes—
143C22×D4—1062no—
144C22×D4—1061no—
145AGL(1,ℤ/8)—2124no—
146AGL(1,ℤ/8)—2124no—
147C2×SD16—1064no—
148C2×D8—1064no—
149C2×M16—22no—
150C2×M16—1061no—
151C22×C8—22no—
152C4⋊Q8C4×D5311yescharacteristic
153#127—532no—
154#127—1061no—
155#127—1062no—
156#127C4×D5311yescharacteristic
157#124D21212yes—
158#121—22no—
159C4×D4—22no—
160C2×C42C22×D5311yescharacteristic
161C4⋊C8—1062no—
162C4⋊C8C22⋊D5314yes—
163C22⋊C8—24no—
164C22⋊C8—22no—
165C22⋊C8—1062no—
166#97—22no—
167#97—1061no—
168#97—1062no—
169C4×C8—532no—
170C24—2121no—
171#41—42no—
172#41—42no—
173C2×Q8#1171992112yes—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—1062no—
177C2×D4—1064no—
178C2×D4—4244no—
179C2×D4—2122no—
180C2×D4—2122no—
181C2×D4—2121no—
182C2×D4—2122no—
183C2×D4#1171992112yes—
184C22×C4—22no—
185C22×C4#1171994512yes—
186C22×C4C2×C22⋊D5311yescharacteristic
187SD16—2128no—
188D8—2128no—
189M16—82no—
190M16—1064no—
191C2×C8—24no—
192C2×C8—82no—
193C2×C8—1062no—
194C2×C8—2122no—
195C4⋊C4—22no—
196C22⋊C4—24no—
197C42C2×C4×D5311yescharacteristic
198C42C2×D21211yescharacteristic
199C42C2×C22⋊D5312yes—
200C23—22no—
201C23—4244no—
202C23—4241no—
203C23—4241no—
204C23—2122no—
205C23#1313320011yescharacteristic
206Q8—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—4244no—
211D4—4244no—
212D4—4242no—
213D4—2124no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4C2×#1171992111yescharacteristic
220C2×C4C4×C22⋊D5312yes—
221C2×C4#1313318611yescharacteristic
222C2×C4#1313319112yes—
223C8—44no—
224C8—1064no—
225C8—4242no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—44no—
230C22—8482no—
231C22—8482no—
232C22—4242no—
233C22—4244no—
234C22#1681050211yescharacteristic
235C4—21no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4#1681079112yes—
240C2—41no—
241C2—82no—
242C2—8482no—
243C2#2027511311yescharacteristic
244C2#2027519512yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2980C2
socleC22×C53#386
prime radical#20285168C2
cosocleC2×C42×C53C23
hypercenter#2980D53
hypocenterC53#9423
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20273428C2semidirect
3. #20273350C2semidirect
4. #20272586C2semidirect
5. C53×#2980C2semidirect
119. #2980D53semidirect
141. C53#9423semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202851682
#202734282
#202733502
#202725862
C53×#29802
#942353